Okay so I think I have a very trivial and short proof for the change of base rule but I'm worried it might be circular or wrong since I see it nowhere.
To prove $$\log_bx = \frac{\log_ax}{\log_ab}$$
Proving that $\log_ax = \log_ab*\log_bx$ will prove the above since it's the result of multiplying both sides by $\log_ab$ and hence the same relationship.
$\log_ab*\log_bx = \log_ab^{\log_bx}$ // Logarithm power rule.
$\log_ab^{\log_bx} = \log_ax$ // By definition $b^{\log_bx} = x$
Q.E.D. ?